Analysis Study Seminar -- Fall 2006
Thursdays 3:00-5:00 -- East Hall 2866

PAST SEMINARS: WINTER 06

Thursday, December 14.

Speaker: Zair Ibragimov, UofM.
Title: On a theorem of Borsuk and Bott

Abstract: We discuss the notion of symmetric products of topological spaces introduced by Borsuk and Ulam in 1930. For $n\geq 1$, the $n$-th symmetric product of a topological space $X$ is the space of all subsets of $X$ of cordinality less than or equal to $n$ equipped with the quotient topology coming from the product space $\prod_{1}^{n}X$. Our primary focus will be on Bott's proof of the Borsuk-Bott Theorem, which says that the third symmetric product $M$ of a circle $S^1$ is homeomorphic to $S^3$. We also give an alternative proof based on the Poincare' Conjecture. If time permits, we will discuss metric properties of $(M, d)$, where $d$ is the Hausdorff metric induced by the Euclidean metric on $S^1$.

Speaker: Mario Bonk, UofM.
Title: Asymptotic cones (continued)

Abstract: Asymptotic cones are useful in studying the large scale behavior of a metric space. The definition of this concept is based on ultrafilters and ultralimits. I will discuss the basic definitions and present some applications.

Speaker: Mario Bonk, UofM.
Title: Asymptotic cones (continued)

Abstract: Asymptotic cones are useful in studying the large scale behavior of a metric space. The definition of this concept is based on ultrafilters and ultralimits. I will discuss the basic definitions and present some applications.

Thursday, November 23.

Thanksgiving break.

Speaker: Mario Bonk, UofM.
Title: Asymptotic cones

Abstract: Asymptotic cones are useful in studying the large scale behavior of a metric space. The definition of this concept is based on ultrafilters and ultralimits. I will discuss the basic definitions and present some applications.

Speaker: Jang-Mei Wu, University of Illinois.
Title: Tug of war with noise: a game theoretic view of the p-Laplacian ( after Peres and Sheffield) (continued)

Abstract: We study a recent preprint (August 2006) of Y. Peres and S. Sheffield, in which they develop an interpretation of the solutions to the boundary value problem for the p-Laplacian from game theoretic point of view. This can be regarded as an analogue of the Brownian motion for nonlinear equations.

Speaker: Jang-Mei Wu, University of Illinois.
Title: Tug of war with noise: a game theoretic view of the p-Laplacian ( after Peres and Sheffield)

Abstract: We study a recent preprint (August 2006) of Y. Peres and S. Sheffield, in which they develop an interpretation of the solutions to the boundary value problem for the p-Laplacian from game theoretic point of view. This can be regarded as an analogue of the Brownian motion for nonlinear equations.

Thursday, October 26, 2006.

Speaker: Daniel Meyer, UofM.
Title: Moore's theorem (continued)

Abstract: We explain and prove Moores' theorem, which gives a condition when a decomposition space (identification of compact sets) of the sphere yields again the sphere.

Thursday, October 19, 2006.

Speaker: Daniel Meyer, UofM.
Title: Moore's theorem

Abstract: We explain and prove Moores' theorem, which gives a condition when a decomposition space (identification of compact sets) of the sphere yields again the sphere.

Speaker: Pekka Pankka, UofM.
Title: Volume growth and hyperbolicity (continued)

Abstract: I will discuss a theorem of Varopoulos on the growth bounds for fundamental groups of closed quasiregularly elliptic manifolds [Varopoulos - Saloff-Coste - Coulhon, Analysis and geometry on groups, Theorem X.5.1].

Speaker: Pekka Pankka, UofM.
Title: Volume growth and hyperbolicity (continued)

Abstract: I will discuss a theorem of Varopoulos on the growth bounds for fundamental groups of closed quasiregularly elliptic manifolds [Varopoulos - Saloff-Coste - Coulhon, Analysis and geometry on groups, Theorem X.5.1].

Thursday, September 28, 2006.

Speaker: Juha Heinonen, UofM.

Title: Extending Lipschitz functions via random metric partitions (after Lee and Naor)
(continued)

Abstract: I will discuss recent work by J. Lee and A. Naor (Inventiones 2005), where they construct Lipschitz extensions of functions from doubling subsets of arbitrary metric spaces into Banach spaces by using ``random partitions of unity".

Speaker: Juha Heinonen, UofM.

Title: Extending Lipschitz functions via random metric partitions (after Lee and Naor)
(continued)

Abstract: I will discuss recent work by J. Lee and A. Naor (Inventiones 2005), where they construct Lipschitz extensions of functions from doubling subsets of arbitrary metric spaces into Banach spaces by using ``random partitions of unity".

Speaker: Juha Heinonen, UofM.

Title: Extending Lipschitz functions via random metric partitions (after Lee and Naor)

Abstract: I will discuss recent work by J. Lee and A. Naor (Inventiones 2005), where they construct Lipschitz extensions of functions from doubling subsets of arbitrary metric spaces into Banach spaces by using ``random partitions of unity".